Attempting to define a directed version of distance-hereditary graphs, it is necessary to decide which of these definitions modified to a directed definition is most promising to give an useful digraph class. We consider this matter concerning some directed graph parameters.
In [1]}, the authors use a straightforward way in generalizing the property of distance-hereditary, i.e. the property that some shortest path has to be induced subgraph, on digraphs. But their graphs are limited to oriented digraphs,which are digraphs without bidirectional edges. For undirected distance-hereditary graphs, tree-width is computable in linear time [2]}. Further, linear rank-width of distance-hereditary graphs is computable in polynomial time [3]}. The clique-width of any distance-hereditary graph is at most 3 [4]}, but path-width is hard even on bipartite distance-hereditary graphs [5]}.
